From October 1992 to October 1995, 480 photometric brightness measurements of
the well-known eclipsing binary Beta Lyrae were made using a solid-state
photometer along with U, B, V, R, and I filters. Five times of minimum were
graphically determined from the photometric data. Analysis of the light curves
showed that Beta Lyrae was significantly redder during the primary eclipses.
The calculated rate of mass transfer between the stars was 2.7 x
1025 kg/year.

I. INTRODUCTION

Beta Lyrae is an eclipsing binary star that is apparently evolving. New
observations are always needed to help us better understand this dynamical
system. Beta Lyrae's brightness and relatively short period make it a good
target for student observers and beginning photometrists. In this paper we
will show the results of three years of observations by students at the Texas
A&M Observatory.

Beta Lyrae is one of the most familiar, yet oddest variable stars in the sky.
The primary star of this close binary is classified from its spectrum as a
giant of type B7 and probably has about twice the mass of the Sun. The
secondary star seems to be largely invisible even though it appears to be far
more massive with about 12 times the Sun's mass. The secondary is probably
concealed at least partially by a thick, opaque torus of matter orbiting and
spiraling into it. Figure 1 shows a model of the Beta Lyrae based roughly on
modeling by Wilson (1973; 1981).

Figure 1: Model of Beta Lyrae made using Mathematica.

Even the Voyager spacecraft and telescopes onboard the space shuttle have
observed this eclipsing binary star (Kondo 1994; Nordsieck 1995). Observations
from last spring's space shuttle-based Astro-2 mission have revealed some
interesting information about the material around the secondary star. Data
from the polarimeter onboard Astro-2 showed an abrupt shift in the polarization
angle between Beta Lyrae's ultraviolet and visible-light emissions which seems
to imply that there is outlying matter at right angles to the system's
accretion disk. These observations not only give information about the
structure of the material around the secondary but also give the orientation of
the accretion disk rotation axis with respect to celestial north (about 15
degrees measured from celestial north, clockwise looking out from Earth).
Measuring this orbital parameter is extraordinary since this parameter cannot
be determined by modeling eclipsing binary star light curves.

The orbital period of Beta Lyrae has increased from 12.89 days when it was
first observed by John Goodricke in 1784 to a current cycle of 12.94 days
(Gilman 1978). The orbital slowdown must be caused by the lighter primary
losing gas to the secondary - thereby creating the accretion torus. The
changes in the torus and masses of the stars effect the light curves noticeably
over time. Monitoring the subtle changes in the light curve will allow us to
understand the evolution of this and other eclipsing binary stars.

II. PHOTOMETRY

A 14-inch Schmidt-Cassegrain (f/11) telescope was used along with two SSP-3
Optec solid state photometers and filters closely matching the Johnson U, B, V
R and I filter system to measure the brightness of Beta Lyrae. All
measurements were made at the Texas A&M University Observatory located just
west of College Station, Texas (30.6deg.N, 96.3deg.W). The comparison star
used for all measurements was Gamma Lyrae. The aperture of the photometer was
small enough to exclude the light from the visual companion
Beta2 Lyrae which is only 46 arc seconds away. From 1992 to
1995, 480 photometric brightness measurements were made. An Optec SSPCARD (IBM
PC Interface Card) was added to the teaching observatory in 1995 and have made
data collection process much more efficient.

III. RESULTS

Figures 2 through 6 show the U, B, V, R and I light curves (differential
magnitude versus orbital phase). The data were corrected for atmospheric
extinction and the error of each measurement was computed using the method
described by Iacovone (1995) with the average of these error values being +/-
0.016 magnitudes. During primary minimum (phase=0.0) Beta Lyrae became on
average 1.2 magnitudes dimmer in the U filter, 1.1 magnitudes dimmer in the B
filter, 1.0 magnitudes dimmer in the V and R filters and 0.8 magnitudes dimmer
in the I filter. These results show that the Beta Lyrae system becomes redder
during primary minimum. No such color change, however, was observed for the
secondary minimum. Also based on Figures 2 through 6 it appears that there is
some orbit-to-orbit variations in the brightness of Beta Lyrae. This kind of
variation appears to have been seen by others as well (Aslan, 1987; Landis
1973).

Figure 2: Ultraviolet Filter Light Curve.

Figure 3: Blue Filter Light Curve.

Figure 4: Visual Filter Light Curve.

Figure 5: Red Filter Light Curve.

Figure 6: Infrared Filter Light Curve.

Figure 7: Primary Minimum on June 15, 1995 with a Parabolic Fit.

The phase of the orbit was calculated using Harmanec's (1993) ephemeris

T = 2408247.966 + 12.913780 E +
0.00000387196 E2

(1)

which is based on 100 years of observations. This equation gives the
heliocentric Julian date, T, of primary minima on epoch E. Photometry from
Wilson (1974) and Landis (1973) seems to show that at least sometimes the
primary minimum occurs to the right of zero phase. On July 16, 1995 we had the
opportunity to observe Beta Lyrae near zero phase as shown in Figure 7. Note
that strictly speaking the primary minimum occurred at around 0.025 phase on
this day. In other words, the light curve appears not to be symmetric about
zero phase. We believe that features like this and the orbit-to-orbit
variations are due to irregularities in the accretion torus, but the proof will
require more study and observations.

Based on our other photometric observations we were also able to graphically
determine five heliocentric Julian date (HJD) times of primary and secondary
minimum: 2448919.65, 2449184.75, 2449197.75, 2449883.50, 2449889.75. Figure 8
shows the observed minus computed (O-C) diagram for these times of minima along
with observations from the British Astronomical Association (Isles, 1994). The
O-C values were calculated from the old 1993 ephemeris published by Cracow
Observatory, Poland:

T = 2448990.54 + 12.93784 E

(2)

We also observed Beta Lyrae with the naked eye and with the eye through
binoculars using Gamma Lyrae and Zeta Lyrae as comparison stars. Figure 9
shows 159 visual magnitude estimates with the phase again calculated using
Equation 1. The visual data show that the primary minimum is about 0.8
magnitudes deep which is similar to what the photometric data show in the V
filter. These eye observations were part of a campaign organized by Isles
(1993; 1994).

Figure 8: O-C deviations for primary ()
and secondary ()
minima.
The solid symbols are from values in the text, the open symbols are
from BAA observers and the curve is from Equation 1.

Figure 9: Human Eye Magnitude Estimates in 1993 and 1995.

IV. MASS TRANSFER

The rate of mass transfer from the primary to the secondary can be
approximated using the orbital period increase given in Equation 1 and some
elementary physics. The angular momentum of the system about its center of
mass is

(3)

where I is the moment of inertia and is the angular velocity about
the center of mass. For a period, , semimajor axis length, a, and
masses m1 and m2,

(4)

for a circular orbit. Using Kepler's Third Law

(5)

and Equations 3 and 4 we get

(6)

where G is the universal gravitational constant. For the case of conservation
of angular momentum (dL/dt = 0) and conservation of mass (dm1/dt = -dm2/dt),
the time derivative of Equation 6 gives

(7)

From Equation 1, we see that for Beta Lyrae the current instantaneous values
for the period and its derivative are = dT/dE ~ 12.94 days and
d/dt ~ 2.2 x 10-4 days/year = 19 seconds/year. If we use the
accepted values for the masses of each star (m1=2 MSun and m2=12 MSun), then we
find that dm1/dt ~ -2.7 x 1025 kg/year. This corresponds to a mass
transfer of about 4.5 Earth masses per year!

V. CONCLUSIONS

An extensive photometric study of Beta Lyrae has been carried out in the
Johnson U, B, V, R and I filters. During primary minimum we found that Beta
Lyrae became a little redder. Primary minima were found to occur on HJD
2449184.75, 2449197.75, 2449883.50, and secondary minima were found to occur on
HJD 2448919.65 and 2449889.75. The measured slowdown in the orbital period of
Beta Lyrae over the last 100 years has been used to calculate a mass transfer
rate of 2.7 x 1025 kg/year from the primary to the secondary star.

VI. ACKNOWLEDGMENTS

We would like to thank to Phil McJunkins, John Harper and Dan Carona for their
assistance with data collection and analysis. The authors would also like to
thank the Department of Physics for their support of the Texas A&M
Observatory and student projects like this. We are grateful to the Centre de
Données Astronomiques de Strasbourg for access to the SIMBAD database
(http://cdsweb.u-strasbg.fr/).

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